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A generalization of Franklin's partition identity and a Beck-type companion identity

Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, Ren Watson

Abstract

Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.

A generalization of Franklin's partition identity and a Beck-type companion identity

Abstract

Euler's classic partition identity states that the number of partitions of into odd parts equals the number of partitions of into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.

Paper Structure

This paper contains 4 sections, 4 theorems, 70 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Generating Functions
  3. Combinatorial Proof of Theorem \ref{['OD24']}
  4. A Beck-Type Companion Identity for Theorem \ref{['OD24']}

Key Result

Theorem 1.1

For all integers $n,j \geqslant 0$, and $k,b \geqslant 1$,

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof : Proof of Theorem \ref{['OD24']} via generating functions
  • proof : Bijective proof of Theorem \ref{['OD24']}
  • proof : Proof of Theorem \ref{['Ojkb/Djkb-beck']} via generating functions.
  • Proposition 4.1
  • proof
  • proof : Proof of Theorem \ref{['Ojkb/Djkb-beck']} via a modular refinement